Subject: Paper published by Geometry and Topology Originator: edgar@math.ohio-state.edu (Gerald Edgar) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol8/paper25.abs.html Title: Invariants for Lagrangian tori Author(s): Ronald Fintushel, Ronald J Stern Abstract: We deŽne an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have inŽnitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-inŽnity invariant. In addition, this invariant is used to show that many symplectic 4-manifolds have nontrivial homology classes which are represented by inŽnitely many pairwise inequivalent Lagrangian tori, a result Žrst proved by S Vidussi for the homotopy K3-surface obtained from knot surgery using the trefoil knot in [Lagrangian surfaces in a Žxed homology class: existence of knotted Lagrangian tori, J. Diff. Geom. (to appear)]. Secondary: 57R17 Keywords: 4-manifold, Seiberg-Witten invariant, symplectic, Lagrangian Proposed: Peter Kronheimer Seconded: Robion Kirby, Yasha Eliashberg Author(s) address(es): Department of Mathematics, Michigan State University East Lansing, Michigan 48824, USA and Department of Mathematics, University of California Irvine, California 92697, USA Email: ronŽnt@math.msu.edu, rstern@math.uci.edu === Subject: Random Voronoi diagrams Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Imagine dots scattered about a plane according to a Poisson process with uniform intensity. If I¹m not mistaken, and argument using the Euler characteristic shows that the expected number of sides in each Voronoi cell is six. Is any decent expository account in print that explains this and such matters as * The probability distribution of the number of sides of each Voronoi cell; * The probability distribution of the size and shape of each Voronoi cell; * The nature of the dependence in the joint probability distribution of nearby Voronoi cells. Mike Hardy === Subject: Re: Random Voronoi diagrams Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Imagine dots scattered about a plane according to a Poisson >process with uniform intensity. If I¹m not mistaken, and >argument using the Euler characteristic shows that the >expected number of sides in each Voronoi cell is six. >Is any decent expository account in print that explains >this and such matters as Spatial Tessellations: Concepts and Applications of Voronoi Diagrams http://okabe.t.u-tokyo.ac.jp/okabelab/Voronoi/ === Subject: Re: Wiles¹ Proof of Fermat¹s Last Theorem, and n=2 Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Does anyone know of any publications regarding >how Wiles¹ Proof (or related modiŽcations or >extensions) of Fermat¹s Last Theorem behaves when n=2? >If I recall correctly, Wiles¹ Proof speciŽcally begins >with the condition n>=5. Since n=3 and n=4 are >separately proven, this condition n>=5 is sufŽcient >to prove FLT for all prime n <> 2. >I am interested in how Wiles¹ Proof fails for n=2, >to allow the Pythagorean Theorem to hold. >Does the Taniyama-Shimura (-Weil) Conjecture fail? >How does the Frey ellipical curve behave for n=2? >Do the modular forms corresponding to n=2 reduce >to simple forms? I am not an expert, but since no expert has responded, here, for what it¹s worth, is my understanding: To get Fermat¹s Last Theorem, you need to combine Wiles¹s proof that the Frey curve is modular with Ribet¹s proof that the Frey curve is not modular (from which it follows that the Frey curve, along with the alleged FLT counterexample from which it arises, cannot exist). The part that fails for n=2 and n=3 is not the Wiles part but the Ribet part. That¹s because Ribet¹s argument requires that the action of G(Q-bar,Q) on the n-torsion in the Frey curve has to be irreducible. For this, one invokes a theorem of Mazur that requires n > 4. I can explain almost nothing about how Mazur uses that assumption. I do not know whether the Frey curves for n=2 are well understood in general, but they are certainly understood in particular cases. For example, the Frey curve associated to the equation 3^2 + 4^2 = 5^2 is the modular curve X_0(15). I am sure that there is some clear intuition due to Frey about why the Frey curves should be modular for n=2 and n=3 but not modular for n equal to a prime greater than 3. Here¹s where we need that expert. Steven E. Landsburg www.landsburg.com/about2.html >Has there been any discussion on this topic? >Anthony Natoli -- Steven E. Landsburg http://www.landsburg.com/about2.html === Subject: Game theory - theory of promises? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) A question about game theory: In a classic n-player simultaneous move matrix game (eg, the Prisoner¹s dilemma) better outcomes can be obtained if you have some way of making a binding (or believable) promise that you will choose a particular action. Eg: Say you and I are about to play Prisoner¹s Dilemma. But we are given the opportunity to communicate before the game, and I swear upon my life that I will cooperate, if you promise to do likewise. Now it is in your best interest to agree. We both promise to cooperate, and as people of our word, we both maintain our promises. For each of us the payoff is better than it could have been without having made the promises before the game. Alternatively, promises can be regarded as threats. Consider the game of Chicken: YOU Drive Swerve ME Drive (0,0) (3,1) Swerve (1,3) (2,2) There are two equilibria for this game, either combination of Drive/Swerve or Swerve/Drive. If I can make a binding promise to you, before the game, that I will not swerve (perhaps by throwing my steering wheel out the window) then your only rational decision is to swerve. So I guarantee a favourable outcome for me by making an appropriate promise/threat. My question is: has anyone done a full analysis of this kind of promise-making? It seems fairly elementary to me. I¹m interested in doing some research in this area, but I¹m rather new to game-theory. I¹m familiar with most of the standard text-book stuff, but text-books are always behind the times. Is anyone out there up with the state of play? Are there any papers I ought to read? Malcolm === Subject: Re: Logic question - distributing forall and exists over each other? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Problem solved. > forall x in X , exists y in Y_x , P(x,y) > iff > exists f in F , forall x in X , P(x,f(x)) > where F is the set of functions of the form f:X -> U{Y_x|x in X} > such that f(x) in Y_x. It *is* the axiom of choice (in one of its formulations). Good luck, Malcohol. === Subject: Polarized Abelian Varieties Originator: bergv@math.uiuc.edu (Maarten Bergvelt) In Milne¹s online notes on Abelian Varieties, he states in chapter 10: As Weil pointed out, for many purposes, the correct higher dimensional analogue of an elliptic curve is not an abelian variety, but a polarized abelian variety. Alas, the chapter is short, and I still can¹t see why the above statement is true. What is the role of the polarization map? And what if we restrict ourselves to principally polarized abelian varieties? I¹m interested in this because Shimura curves can be interpreted as moduli spaces of certain principally polarized abelian surfaces (speciŽcally, those which have complex multiplication by orders in a quaternion algebra). === Subject: This week in the mathematics arXiv (21 Jun - 25 Jun) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Here are this week¹s titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modiŽcation. Titles in the mathematics arXiv (21 Jun - 25 Jun) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0406415 Anthony J. Crachiola: On automorphisms of Danielewski surfaces math.AC/0406414 Anthony J. Crachiola: Return of $x + x^2y + z^2 + t^3 = 0$ math.AC/0406412 Anthony J. Crachiola, Leonid Makar-Limanov: On the rigidity of small domains math.AC/0406409 Uwe Nagel, Tim Roemer: Extended degree functions and monomial modules math.AC/0406385 Hagen Knaf: Regular local algebras over a Pruefer domain: weak dimension and regular sequences math.AC/0406359 Carlos D¹Andrea, Martin Sombra: The Cayley-Menger determinant is irreducible for $ngeq 3$ math.AC/0406357 Anurag K. Singh, Irena Swanson: Associated primes of local cohomology modules and of Frobenius powers math.AC/0406356 Anurag K. Singh: Associated primes of local cohomology modules math.AC/0406355 Anurag K. Singh: p-torsion elements in local cohomology modules. II math.AC/0406354 Anurag K. Singh: p-torsion elements in local cohomology modules AG: Algebraic Geometry ---------------------- math.AG/0406501 Suresh Nayak: Pasting pseudofunctors math.AG/0406497 Ph. Ellia, D. Franco: On Smooth Divisors of a Projective Hypersurface math.AG/0406494 Ciro Ciliberto, Francesco Russo: Varieties with minimal secant degree and linear systems of maximal dimension on surfaces math.AG/0406493 Ivan V. Arzhantsev, Natalia A. Tennova: On afŽnely closed homogeneous spaces math.AG/0406410 Jaydeep V. Chipalkatti: On the invariant theory of the Bezoutiant math.AG/0406396 H¹el`ene Esnault: Variation on Artin¹s vanishing theorem math.AG/0406390 Misha Verbitsky: Hypercomplex structures on Kaehler manifolds math.AG/0406386 Dragutin Svrtan, Igor Urbiha: Atiyah-Sutcliffe Conjectures for Almost Collinear ConŽgurations and Some New Conjectures for Symmetric Functions math.AG/0406384 Mike Roth, Ravi Vakil: The afŽne stratiŽcation number and the moduli space of curves math.AG/0406383 Laura Felicia Matusevich, Ezra Miller, Uli Walther: Homological Methods for Hypergeometric Families math.AG/0406380 Tamas Hausel: Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve AP: Analysis of PDEs -------------------- math.AP/0406495 Tonia Ricciardi: A sharp Holder estimate for elliptic equations in two variables nlin.SI/0406038 Henrik Aratyn, Johan van de Leur: The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlev¹e VI nlin.CD/0406049 Dhrubaditya Mitra, Jeremie Bec, Rahul Pandit, Uriel Frisch: Multiscaling in the Stochastically Forced Burgers Equation math.AP/0406403 J. Douglas Wright: Corrections to the KdV approximation for water waves math.AP/0406368 Haakan Hedenmalm, Anders Olofsson: Hele-Shaw žow on weakly hyperbolic surfaces math.AP/0406362 Eric Gautier: Large deviations and support results for nonlinear Schrodinger equations with additive noise and applications AT: Algebraic Topology ---------------------- math.AT/0406502 Tornike Kadeishvili: On the cobar construction of a bialgebra math.AT/0406483 J.F. Jardine: Fibred sites and stack cohomology math.AT/0406405 Peter Bubenik: Free cell attachments and the rational homotopy Lie algebra math.AT/0406363 Imma Galvez, Sarah Whitehouse: InŽnite sums of Adams operations and cobordism math.AT/0406361 Michael Farber: Collision Free Motion Planning on Graphs CA: Classical Analysis and ODEs ------------------------------- math.CA/0406489 Victor Katsnelson, Dan Volok: Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions II math.CA/0406484 K. T.-R. McLaughlin, P. D. Miller: The dbar steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with Žxed and exponentially varying nonanalytic weights math.CA/0406401 P. Freitas: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums math.CA/0406375 Yuval Peres, Boris Solomyak: The sharp Hausdorff measure condition for length of projections math.CA/0406372 Erik Talvila: Estimates of the remainder in Taylor¹s theorem using the Henstock--Kurzweil integral math.CA/0406371 Erik Talvila: Estimates of Henstock--Kurzweil Poisson integrals math.CA/0406370 Peter A. Loeb; Erik Talvila: Lusin¹s Theorem and Bochner Integration CO: Combinatorics ----------------- quant-ph/0406165 S. L. Braunstein, S. Ghosh, S. Severini: The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states math.CO/0406485 Christian Haase, Ilarion V. Melnikov: The režexive dimension of a lattice polytope math.CO/0406450 Andrew Rechnitzer: Haruspicy 2: The self-avoiding polygon generating function is not D- Žnite math.CO/0406420 Leonid Gurvits: Van der Waerden Conjecture for Mixed Discriminants math.CO/0406418 Marcelo Aguiar, Kathryn Nyman, Rosa Orellana: New results on the peak algebra math.CO/0406381 David callan: Bijections from restricted Dyck paths to Motzkin paths quant-ph/0406135 Sudhir Kumar Singh, Sudebkumar Prasant Pal, Somesh Kumar, R. Srikanth: A combinatorial approach for studying LOCC transformations of multipartite states math.CO/0406378 Hyuk Han, Seunghyun Seo: Combinatorial proofs of inverse relations and log-concavity for Bessel numbers math.CO/0406374 Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor: On Metric Ramsey-type Dichotomies math.CO/0406373 Nicholas J. Proudfoot: Hyperplane arrangements and K-theory CT: Category Theory ------------------- math.CT/0406475 Josep Elgueta: A strict totally coordinatized version of Kapranov and Voevodsky¹s 2-category {bf 2Vect} CV: Complex Variables --------------------- math.CV/0406500 V. H. Jorge Perez: Counting isolated singularities in germs of applications ${C^n,0to C^p,0}$, n

Toshikazu Ito, Bruno Scardua: On non-integrable complex distributions: an approach by transversality with real domains math.CV/0406408 Leon A. Takhtajan, Lee-Peng Teo: Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping math.CV/0406387 G. Dloussky: On surfaces of class VII_0^+ with numerically anticanonical divisor math.CV/0406376 C.P. Hughes, A. Nikeghbali: The zeros of random polynomials cluster uniformly near the unit circle math.CV/0406369 Alexandru Aleman, Haakan Hedenmalm, Stefan Richter, Carl Sundberg: Curious properties of canonical divisors in weighted Bergman spaces DG: Differential Geometry ------------------------- math.DG/0406492 Andrei Moroianu, Paul-Andi Nagy, Uwe Semmelmann: Unit Killing Vector Fields on Nearly Kahler Manifolds math.DG/0406490 Christopher Allday: Canonical equivariant extensions using classical Hodge theory math.DG/0406445 Martin Bojowald, Alexei Kotov, Thomas Strobl: Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries math.DG/0406439 Jeanne N. Clelland, Christopher G. Moseley: Sub-Finsler geometry in dimension three math.DG/0406437 Jun Ling: The First Closed Eigenvalue and the Li Conjecture math.DG/0406428 Nikolaos I. Katzourakis: The Riemann Surface of the Logarithm Constructed in a Geometrical Framework math.DG/0406426 Benoit Daniel: Isometric immersions into S^n x R and H^n x R and applications to minimal surfaces math.DG/0406422 Chuu-Lian Terng, Erxiao Wang: Curved žats, exterior differential systems, and conservation laws math.DG/0406400 Pawel Nurowski: Differential equations and conformal structures math.DG/0406399 Brendan Guilfoyle, Wilhelm Klingenberg: The Casimir Effect Between Non-Parallel Plates by Geometric Optics math.DG/0406397 Anton S. Galaev: Remark on holonomy groups of pseudo-Riemannian manifolds of signature (2,n+2) math.DG/0406393 Sergiu I. Vacaru: Nonlinear Connections and Exact Solutions in Einstein and Extra Dimension Gravity DS: Dynamical Systems --------------------- math.DS/0406503 Richard Kenyon, Lorenzo Sadun, Boris Solomyak: Topological mixing for substitutions on two letters math.DS/0406457 V.M. Buchstaber, S.Yu. Shorina: $w$-function of the KdV hierarchy math.DS/0406442 Jana Rodriguez Hertz: Continuum-wise expansive homeomorphisms on Peano continua math.DS/0406417 Juan Rivera-Letelier: Wild recurrent critical points math.DS/0406416 Mark Braverman, Michael Yampolsky: Non-computable Julia sets math.DS/0406413 I. D. Shkredov: On Multiple Recurrence math.DS/0406367 Tien-Cuong Dinh, Nessim Sibony: Dynamics of regular birational maps in P^k math.DS/0406360 Nikos Frantzikinakis, Bryna Kra: Convergence of multiple ergodic averages for some commuting transformations FA: Functional Analysis ----------------------- math.FA/0406479 Valentin Ferenczi, Eloi Medina Galego: Some results about the Schroeder-Bernstein Property for separable Banach spaces math.FA/0406477 Valentin Ferenczi, Eloi Medina Galego: Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces math.FA/0406391 E.Ostrovsky, L.Sirota: Fourier tranform in exponential rearrangement invariant spaces GN: General Topology -------------------- math.GN/0406411 Boaz Tsaban: SPM Bulletin 9 GR: Group Theory ---------------- math.GR/0406443 Sean Cleary, Tim R. Riley: A Žnitely presented group with inŽnite dead end depth math.GR/0406382 Anton A. Klyachko: How to generalize known results on equations over groups GT: Geometric Topology ---------------------- math.GT/0406498 A.Pajitnov: Novikov homology, twisted Alexander polynomials and Thurston cones math.GT/0406486 David G. C. Handron: The Morse Complex for a Morse Function on a Manifold with Corners math.GT/0406407 Ian Agol: Transcendental ending laminations math.GT/0406402 Matthew Hedden: On Knot Floer Homology and Cabling math.GT/0406377 Allen Hatcher, Karen Vogtmann: Homology stability for outer automorphism groups of free groups LO: Logic --------- math.LO/0406505 Wesley Calvert: The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length math.LO/0406482 Saharon Shelah: More on: the revised GCH and middle diamond math.LO/0406481 Tapani Hyttinen, Olivier Lessmann, Saharon Shelah: Interpreting groups and Želds in some nonelementary classes math.LO/0406441 Menahem Kojman, Wieslaw Kubis, Saharon Shelah: On two problems of Erdos and Hechler: New methods in singular Madness math.LO/0406440 Saharon Shelah: Dependent Žrst order theories, continued math.LO/0406438 Thomas Jech, Saharon Shelah: Simple Complete Boolean Algebras MG: Metric Geometry ------------------- math.MG/0406448 Sever Silvestru Dragomir: Additive Reverses of the Generalised Triangle Inequality in Normed Spaces math.MG/0406406 Artem Zvavitch: The Busemann-Petty problem for arbitrary measures math.MG/0406404 Yair Batal, Nathan Linial, Manor Mendel, Assaf Naor: Limitations to Frechet¹s Metric Embedding Method math.MG/0406394 Ronald L. Graham, Boris D. Lubachevsky: Repeated Patterns of Dense Packings of Equal Disks in a Square math.MG/0406358 Yair Bartal, Nathan Linial. Manor Mendel, Assaf Naor: On some low distortion metric Ramsey problems math.MG/0406353 Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor: On metric Ramsey-type phenomena MP: Mathematical Physics ------------------------ math-ph/0406061 Edwin Langmann: Addendum to: Second quantization of the elliptic Calogero-Sutherland model math-ph/0406060 V. V. Varlamov: On the CPT group of the Dirac Želd in de Sitter space math-ph/0406059 G. Gonzalez: Lagrangians and Hamiltonians for one-dimensional systems math-ph/0406058 Brice Camus: Semiclassical spectral estimates for Schrodinger operators at a critical energy level math-ph/0406057 Manuel Calixto, Julio Guerrero: Wavelet Transform on the Circle and the Real Line: UniŽed Group-Theoretical Treatment math-ph/0406056 Paolo Maria Mariano: SO(3) invariance and covariance in mixtures of simple bodies math-ph/0406055 A. Fannjiang, S. Nonnenmacher, L. Wolowski: Dissipation Time of Quantized Toral Maps math-ph/0406054 A.W.Beckwith: The tunneling hamiltonian representation of false vacuum decay: III. Application to nucleation of an inžationary universe math-ph/0406053 A.W. Beckwith, J.H. Miller Jr: The tunneling hamiltonian representation of false vaccuum decay: II. Application to soliton - anti soliton pair creation math-ph/0406049 R.K. Saxena, A.M. Mathai, H.J. Haubold: Astrophysical Thermonuclear Functions for Boltzmann-Gibbs Statistics and Tsallis Statistics math-ph/0406047 R.K. Saxena, A.M. Mathai, H.J. Haubold: UniŽed Fractional Kinetic Equation and a Fractional Diffusion Equation math-ph/0406046 R.K. Saxena, A.M. Mathai, H.J. Haubold: On Generalized Fractional Kinetic Equations quant-ph/0406157 Taksu Cheon: Altruistic Contents of Quantum Prisoner¹s Dilemma math-ph/0406052 L. A. Khodarinova, I. A. Prikhodsky: On algebraic integrability of the deformed elliptic Calogero-Moser problem math-ph/0406051 Edwin Langmann: Addendum to: Second quantization of the elliptic Calogero-Sutherland model math-ph/0406050 L. A. Khodarinova, I. A. Prikhodsky: Algebraic spectral relations for elliptic quantum Calogero-Moser problems math-ph/0406048 A.W.Beckwith, J.H.Miller Jr: Tunneling Hamiltonian representation of false vacuum decay I. Comparison with the Bogomil¹nyi inequality math-ph/0406045 Nikolai Filonov, Frederic Klopp: Absolutely continuous spectrum for the isotropic Maxwell operator with coefŽcients that are periodic in some directions and decay in others math-ph/0406044 John Cardy: Network Models in Class C on Arbitrary Graphs math-ph/0406043 M. C. Valsakumar, A. Rajan Nambiar, P. Rameshan: Stability of differential equations associated with a class of one dimensional maps math-ph/0406042 Kazuhiro Hikami, Anatol N. Kirillov: Hypergeometric Generating Function of L-function, Slater¹s Identities, and Quantum Invariant math-ph/0406041 G. A. Hagedorn, A. Joye: Determination of Non-Adiabatic Scattering Wave Functions in a Born-Oppenheimer Model math-ph/0406040 Naomichi Suzuki, Minoru Biyajima: Analytic solution for Brownian motion in three dimensional hyperbolic space math-ph/0406039 G. Gaeta, P. Morando: Variational principles for involutive systems of vector Želds math-ph/0406038 Taichiro Takagi: Inverse scattering method for a soliton cellular automaton math-ph/0406037 G. Gaeta: Lie-Poincare¹ transformations and a reduction criterion in Landau theory math-ph/0406036 Chiara de Fabriitis, Paolo Maria Mariano: Geometry of interactions in complex bodies math-ph/0406035 Chin-Sheng Wu: The Recurrence Relation of Irreducible Tensor Operators for O(4) NT: Number Theory ----------------- math.NT/0406476 Patrice Philippon, Martin Sombra: Normalized height of projective toric varieties math.NT/0406461 Luis Dieulefait, Mladen Dimitrov: Explicit determination of the images of the Galois representations attached to Hilbert modular forms math.NT/0406434 Jesse I. Deutsch: A Quaternionic Proof of the Representation Formula of a Quaternary Quadratic Form math.NT/0406429 Jesse I. Deutsch: A Quaternionic Proof of the Universality of Some Quadratic Forms hep-th/0406113 J.S.Dowker: Spherical Casimir energies and Dedekind sums math.NT/0406388 Andrea Mori: Power series expansions of modular forms and their interpolation properties math.NT/0406366 Yoaka Hachimori, Romyar ShariŽ: On the failure of pseudo-nullity of Iwasawa modules OA: Operator Algebras --------------------- math.OA/0406488 Hari Bercovici: Multiplicative Monotone Convolution math.OA/0406458 D. Gwion Evans: On the K-theory of higher rank graph C*-algebras OC: Optimization and Control ---------------------------- math.OC/0406435 Carlo Marinelli: Optimal distributed dynamic advertising math.OC/0406419 Leonid Gurvits, Leiba Rodman: On Matrix Polynomials with Real Roots PR: Probability --------------- math.PR/0406504 Elchanan Mossel, Ryan O¹Donnell: Coin žipping from a cosmic source: On error correction of truly random bits math.PR/0406487 Manjunath Krishnapur, Yuval Peres: Recurrent graphs where two independent random walks collide Žnitely often math.PR/0406459 Ashkan Nikeghbali, Marc Yor: A deŽnition and some characteristic properties of pseudo-stopping times math.PR/0406447 Svante Janson, Elchanan Mossel: Robust reconstruction on trees is determined by the second eigenvalue math.PR/0406446 Elchanan Mossel: Survey: Information žow on trees math.PR/0406444 Davar Khoshnevisan, Yimin Xiao: Levy Processes: Capacity and Hausdorff Dimension math.PR/0406423 Rainer Siegmund-Schultze, Heinrich von Weizsaecker: Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks math.PR/0406398 M. Kozlova, P. Salminen: Diffusion local time storage math.PR/0406392 Rainer Siegmund-Schultze, Heinrich von Weizsaecker: Level Crossing Probabilities I: One-dimensional Random Walks and Symmetrization math.PR/0406379 Marek Biskup: Graph diameter in long-range percolation math.PR/0406364 Shannon Starr: A Thinning Analogue of de Finetti¹s Theorem QA: Quantum Algebra ------------------- math.QA/0406499 Pavel Etingof: Cherednik and Hecke algebras of varieties with a Žnite group action math.QA/0406480 Pavel Etingof, Alexei Oblomkov, Eric Rains: Generalized double afŽne Hecke algebras of rank 1 and quantized Del Pezzo surfaces math.QA/0406478 Andre Henriques, Joel Kamnitzer: Crystals and coboundary categories math.QA/0406389 James Conant, Karen Vogtmann: Morita classes in the homology of automorphism groups of free groups RA: Rings and Algebras ---------------------- math.RA/0406436 S. Caenepeel, E. De Groot, J. Vercruysse: Galois theory for comatrix corings: descent theory, Morita theory, Frobenius and separability properties math.RA/0406365 Csaba Schneider: A computer-based approach to the classiŽcation of nilpotent Lie algebras SG: Symplectic Geometry ----------------------- math.SG/0406449 Yong-Geun Oh: Floer mini-max theory, the Cerf diagram, and the spectral invariants SP: Spectral Theory ------------------- math.SP/0406496 Colin Guillarmou: Absence of resonance near the critical line on asymptotically hyperbolic spaces math.SP/0406491 Nedelec Laurence: Perturbations of non self-adjoint Sturm-Liouville problems, with applications to harmonic oscillators math.SP/0406395 I. Egorova, L. Golinskii: On location of discrete spectrum for complex Jacobi matrices ST: Statistics -------------- math.ST/0406474 Bradley Efron, Trevor Hastie, Iain Johnstone, Robert Tibshirani: Rejoinder to Least angle regression by Efron et al math.ST/0406473 Sanford Weisberg: Discussion of Least angle regression by Efron et al math.ST/0406472 Berwin A. Turlach: Discussion of Least angle regression by Efron et al math.ST/0406471 Robert A. Stine: Discussion of Least angle regression by Efron et al math.ST/0406470 Saharon Rosset, Ji Zhu: Discussion of Least angle regression by Efron et al math.ST/0406469 David Madigan, Greg Ridgeway: Discussion of Least angle regression by Efron et al math.ST/0406468 Jean-Michel Loubes, Pascal Massart: Discussion of Least angle regression by Efron et al math.ST/0406467 Keith Knight: Discussion of Least angle regression by Efron et al math.ST/0406466 Jianqing Fan, Heng Peng: Nonconcave penalized likelihood with a diverging number of parameters math.ST/0406465 Florentina Bunea: Consistent covariate selection and post model selection inference in semiparametric regression math.ST/0406464 Maria Maddalena Barbieri, James O. Berger: Optimal predictive model selection math.ST/0406463 Hemant Ishwaran: Discussion of Least angle regression by Efron et al math.ST/0406462 Peter C. B. Phillips, Katsumi Shimotsu: Local Whittle estimation in nonstationary and unit root cases math.ST/0406460 James O. Berger, Luis R. Pericchi: Training samples in objective Bayesian model selection math.ST/0406456 Bradley Efron, Trevor Hastie, Iain Johnstone, Robert Tibshirani: Least Angle Regression math.ST/0406455 Kalyan Das, Jiming Jiang, J. N. K. Rao: Mean squared error of empirical predictor math.ST/0406454 Galin L. Jones, James P. Hobert: SufŽcient burn-in for Gibbs samplers for a hierarchical random effects model math.ST/0406453 Jae Kwang Kim: Finite sample properties of multiple imputation estimators math.ST/0406452 Bin Nan, Mary J. Emond, Jon A. Wellner: Information bounds for Cox regression models with missing data math.ST/0406451 Guobing Lu, John B. Copas: Missing at random, likelihood ignorability and model completeness math.ST/0406433 Ching-Kang Ing: Selecting optimal multistep predictors for autoregressive processes of unknown order math.ST/0406432 Istv¹an Berkes, Lajos Horv¹ath: The efŽciency of the estimators of the parameters in GARCH processes math.ST/0406431 Anton Schick, Wolfgang Wefelmeyer: Estimating invariant laws of linear processes by U-statistics math.ST/0406430 Danilo Mercurio, Vladimir Spokoiny: Statistical inference for time-inhomogeneous volatility models math.ST/0406427 T. Tony Cai, Mark G. Low: Minimax estimation of linear functionals over nonconvex parameter spaces math.ST/0406425 Yannick Baraud: ConŽdence balls in Gaussian regression math.ST/0406424 Eric D. Kolaczyk, Robert D. Nowak: Multiscale likelihood analysis and complexity penalized estimation -- / Greg Kuperberg (UC Davis) / / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that¹s Žt to e-print * === Subject: Re: open problem? divisors and antichains Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Before comupting the average order of g(n), let me ask if a formula > for g(n) is known in terms of the factorization of n into prime > powers. > TIA Jadek Hello there, I do not know of a formula, but I can paraphrase the deŽnition in case it is of help. My Perl program and my plot use the idea explained below. Suppose n = p1^v1 p2^v2 p3^v3 ... pm^vm and let v = v1 + v2 + v3 + ... + vm s(t) = { (x1, x2, x3, ..., xm) | 0 <= x1 <= v1, 0 <= x2 <= v2, 0 <= x3 <= v3, ..., 0 <= xm <= vm, x1 + x2 + x3 + ... + xm = t } then g(n) = max_{0 <= t <= v} |s(t)|. I hope I explained it correctly. It¹s only a conjecture. Maybe we could get started by proving it or by Žnding a better formula. Now that I think about it we might have g(n) = |s(v/2)|, v even and g(n) = |s((v+1)/2)|, v odd. >The following problem was posted to sci.math and the AofA Bulletin >board but has received no reply yet. Enjoy! >Hi all, >I would like to pose the following problem, which may or may not have >been solved already. >The divisors of a positive integer n form a partially ordered set >(poset). Let g(n) be the size of the largest antichain (set of >pairwise incomparable elements). Compute the asymptotic expansion of >the average order of g(n), i.e. > n > 1 > - > g(k) > n / > k = 1 -- +------------------------------------------------------------+ | Marko Riedel, EDV Neue Arbeit gGmbH, mriedel@neuearbeit.de | | http://www.geocities.com/markoriedelde/index.html | +------------------------------------------------------------+ === Subject: Re: functional analysis and Feigenbaum X-mailer: epigone Epigone-thread: plootwanjul Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>The Cvitanovic-Feigenbaum functional equation >> f(f(a*x)) = a*f(x), 0 < a < 1 >>has inŽnite solutions that can be calculated analytically. The set >>inŽnite solutions found by Alain is one of the possibile inŽnite >>sets of solutions. In order to Žnd a single function f, you need a >>certain number of conditions for describing how the solution behaves >>near its maximum (which is typically the point x=0). >>It may be demonstrated (I did it by myself, but I do not know whether >>someone has ever done it before) that the Cvitanovic-Feigenbaum >>equation can be split into two simpler functional equations >>(relatively) easy to solve. The technique used for simplifying the >>Cvitanovic-Feigenbaum functional equation is disclosed in the paper >>Analytic solutions of the Cvitanovic-Feigenbaum equation, that I >>have submitted for publication to an international journal. You may >>freely download a draft of this paper at this website: >>href=http://media.supereva.it/luigi1132/_private/ EqFeigenbaum.pdf>http: //media.supereva.it/luigi1132/_private/EqFeigenbaum.pdf >>Gaetano Barbaro >I am sure there is still much work to be done for all of us to present >clear tools to solve functional equations. >1) the proposed system {f(y) =g(1+h(y)) and h(g(x))= x } is nothing >else than the very known form f(y) = h^[-1](1+h(y)) where h is a >¹counting¹ function namely h(f^1(y)) = h(y) + 1 ,generalized : >h(f^r(y)) = h(y) + r r being a real. >2)Invariant expressions may easily be built with h( )... >So you limit the set of f functions,few functions are Œcountable¹. >3)my initial solution was a built one;based on Œformal¹ stability: >I¹ve tested it on other cases. I know that considering only solutions of the form f(y) = h^[-1](1+h(y)) is a remarkable limitation, but this ad hoc expedient could be appropriate for calculating solutions of the renormalization equation and also for calculating the Feigenbaum number alpha=2.5029... Solutions of the Cvitanovic-Feigenbaum equation are calculated by Žnding non constant periodic functions A(.) and B(.) with period 1 that satisfy eq. (12) (see my paper). Equation (12) could be satisŽed even if alpha is a positive number. It seems to me that there are inŽnite functions A(.) and B(.) that satisfy eq. (12) for alpha=2.5029, and thus there should be inŽnite solutions of the renormalization equation in the form f(y) = h^[-1](1+h(y)) for alpha=2.5029. Among these inŽnite solutions there could be one function having a quadratic behavior near the origin y=0. Therefore it is not possible to exclude that the function f(y) we are looking for be not a Œcountable¹ function. I have not found any book or paper explaining the meaning of the negative Feigenbaum number alpha=-1/SQRT(2). The book Chaos and nonlinear dynamics by Hilborn, Oxford Press, leaves this question unanswered. Could you suggest an explanation? Gaetano === Subject: 4 x 6 = 24 Originator: baez@math-cl-n03.math.ucr.edu (John Baez) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) There are interesting isomorphisms between these four 24-element groups: Hurwitz integral quaternions of norm 1 = double cover of rotational symmetry group of the tetrahedron = SL(2,Z/3) = deck transformation group of the all-important 24-sheeted branched cover of the moduli space of elliptic curves and these were explained here: http://math.ucr.edu/home/baez/week197.html http://math.ucr.edu/home/baez/week198.html But, I just noticed that they restrict to these other isomorphisms: 4-element cyclic subgroup generated by any fourth root of unity in the Hurwitz integral quaternions (say i) = double cover of rotational symmetries of the tetrahedron that preserve an edge = 4-element cyclic subgroup of SL(2,Z/3) generated by 0 -1 1 0 = rotational symmetries of a square lattice and these: 6-element cyclic subgroup generated by any sixth root of unity in the Hurwitz integral quaternions (say (1 + i + j + k)/2) = double cover of rotational symmetries of the tetrahedron that preserve a vertex = 6-element cyclic subgroup of SL(2,Z/3) generated by 0 -1 1 1 = rotational symmetries of a hexagonal lattice So, something interesting is going on here, and I wonder if anyone knows what it is. === Subject: Re: InŽnite loop spaces and CP_infty Originator: baez@math-cl-n03.math.ucr.edu (John Baez) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >So waaaaaaaaaaayy back in >http://www.math.ucr.edu/home/baez/week149.html in the year we locally >call 2000, John Baez more or less asked for a nice description of what >K(Z,3) looks like; and maybe one appears in later weeks, and maybe >it¹s already been constructed here, but I¹m not quite sure how to go >looking for such a thing. There¹s an interesting description of K(Z,3) waaaaaaaaaayy back in http://math.ucr.edu/home/baez/week151.html , and also more information on K(Z,2), which you may enjoy. >And I also don¹t feel the least bit able to construct such a thing >myself, but I would like to ask what may be a relevant question, since >K(Z,2) is identiŽed as being (homotopic to) CP^infty, either as a >forgotten žag of all Žnite-dimensional CP^n, or as the quotient >space of some Hilbert space H by the natural action of C^* on H. So >just to be absolutely clear, I¹m curious as to which Hilbert space >this is; because, as we learned in Analysis last semester, there¹s a >distict H_kappa with complete ortho-normal system of cardinality >kappa for all cardinals kappa, and that for kappa < lambda, there >is no continuous surjection from H_kappa to H_lambda. >I¹m willing to bet that the H in question is H_{aleph_0}, [...] You¹re right; this countable-dimensional H is the one von Neumann originally called Hilbert space, and this is the one physicists take as the default when they say Hilbert space. >because >CP^{aleph_0} is the space one gets with the above deŽnition >this is maybe interesting because it looks like K(Z,2) is also the set >of quantum states (aleph_0 of them forming a complete ortho-normal You¹re right, this is true. >I can¹t see >yet why K(Z,1) should be considered the set of quantum states of a >think about. Alas, the pattern doesn¹t work this way. But you may enjoy my režections on quantum physics and K(Z,n)¹s in week151 - and you¹ll see that the model given of K(Z,3) also has a lot to do with quantum physics! So, there¹s something going on here which is not fully understood. It must have a lot to do with n-gerbes - see Brylinski¹s book. (Since it¹s taken a long time to reply, I¹ll cc this to you.)